domainCost
plain-language theorem explainer
Domain cost assigns to a mass–energy pair the recognition cost of their ratio: J(m/e). Materials and structural-threshold arguments in RS cite it whenever a dimensionless m/e must be scored by the unique cost forced by the Recognition Composition Law. The body is a one-line specialization of Jcost.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$ is the recognition cost of a positive ratio.
background
Recognition Science fixes a unique cost on positive ratios. The functional $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$) is forced by the Recognition Composition Law and appears as T5 in the forcing chain. Upstream modules record the same definition under the name Jcost and note that a genuine distinction (ratio not one) has strictly positive cost, and that $J$ is nonnegative for $x>0$.
This file is Materials RS Structural Module 9. The module header places it on the chain T5 (J uniqueness) through T8 ($D=3$), and marks the development as a structural theorem block with zero sorry and zero axiom. Domain cost is the local wrapper that feeds mass and energy into that cost as a single ratio.
proof idea
Definitional one-liner: evaluate the shared Jcost functional at the quotient $m/e$. No lemmas are applied; the body is pure abbreviation of the upstream cost.
why it matters
Gives the materials layer a named handle on J-cost of mass-to-energy ratios, so later structural claims (nonnegativity, equality at matched scales, canonical thresholds) can quote a single symbol rather than inline $J(m/e)$. It sits under the T5 landmark: uniqueness of $J$ from the Recognition Composition Law. Sibling results in the same module (domainCost_nonneg, domainCost_at_eq, canonicalThreshold, and the RS_MAT_Structural_009 certificate) are the intended consumers; the used-by graph is still empty, so this is infrastructure rather than a cited lemma yet.
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